On the Stone-Weierstrass theorem in commutative algebra (Q1344584)

The authors' main result is the following: Let \(R\) be a Noetherian ring, and let \(I\) be a proper ideal of \(R\) such that \(R\) is Hausdorff with respect to the \(I\)-adic topology \(\tau_I\). Let \(\widehat {R}\) be the completion of \((R, \tau_I)\), and denote \(C(\widehat {R}, \widehat {R})\) the ring of continuous functions from \(\widehat {R}\) to \(\widehat {R}\) furnished with the topology of uniform convergence. The subset of polynomial functions is dense in \(C(\widehat {R}, \widehat {R})\) if and only if the radical of \(I\) is a maximal ideal \(m\) of \(R\), and the localization \(R_m\) of \(R\) at \(m\) is a one-dimensional analytically irreducible ring with finite residue field.